LES and DNS of symmetrically roughened turbulent channel flows

Abstract

A fully developed turbulent channel flow with symmetrically roughened walls is investigated, where the channel walls are roughened with square ribs, elongated along the span of the channel and are spaced uniformly in the streamwise direction at a constant pitch. The effects of Reynolds number variation on the statistical quantities, the near-wall dynamical structures and on the anisotropic nature of turbulence are studied at two Reynolds numbers ${\text { Re}}_{\tau } =$ 180 and 400, where ${\text{ Re }}_{\tau }$ is based on the channel half-height h and the wall friction velocity $u_{\tau }$. Near-wall resolving large eddy simulations (LES) with different grid resolutions are carried out and validated with in-house direct numerical simulation (DNS) data. Turbulence anisotropy at both small and large scales of motion is investigated using anisotropic invariant maps. A variation in the anisotropic behavior of the flow in the near-wall region is noticed, where the flow is found to be more anisotropic at ${\text{ Re }}_{\tau }=180$ than at ${\text{ Re }}_{\tau }=400$. Also, the anisotropic behavior of the small-scale motions varies from the large-scale motions at ${\text{ Re }}_{\tau }=400$. Two-point correlation and phase analysis using Hilbert transform reveals that the flow within the cavity is independent of the flow outside the cavity. The relatedness of the ‘worm-like’ vortical structures with the positive enstrophy production rate ($\omega _{i}S_{ij}\omega _{j}>0$) is investigated. The regions of positive enstrophy production rate are observed to be topologically ‘sheet-like’ predominantly at a height just above the rib. The regions of negative enstrophy production rate ($\omega _{i}S_{ij}\omega _{j}<0$) are less dominant, with a topology combination of weakly ‘sheet-forming’ and ‘tube-forming’. The statistical features could be captured by LES with a grid consisting of only one-fifth of the total number of grid points as that in the DNS mesh.

Appendix

1.1 Governing equations and subgrid-scale model:

In LES, the filtered quantities representing the larger 3D unsteady turbulent motions are solved directly, whereas the effects of the smaller scale motions are modeled. The filtered quantities are denoted with an overbar. The resolved fields are given by

$$\begin{aligned} {\bar{U}}(x,t)= & {} \int G(r,x;\Delta )U(x-r,t)dr, \end{aligned}$$

(A1)

where $\Delta$ indicates the spatial length scale up to which the scales of motion are resolved. The box filter used in this study is given by

$$\begin{aligned} G_{\Delta }(X-r) = {\left\{ \begin{array}{ll} \frac{1}{\Delta } \quad \mid X-r \mid \le \frac{\Delta }{2}, \\ 0 \quad \quad \, \text {otherwise}. \end{array}\right. } \end{aligned}$$

(A2)

The filtered continuity and momentum equations for LES in the context of an incompressible and isothermal fluid flow take the following form in a Cartesian coordinate system:

$$\begin{aligned} \frac{\partial {\bar{U}}_i}{\partial x_i}= & {} 0, \end{aligned}$$

(A3)

$$\begin{aligned} \frac{\partial {\bar{U}}_i}{\partial t} + \frac{\partial ({\bar{U}}_i {\bar{U}}_j)}{\partial x_j}= & {} -\frac{1}{\rho }\frac{\partial {\bar{P}}}{\partial x_i} + \nu \frac{\partial ^2 {\bar{U}}_i}{\partial x_j \partial x_j} - \frac{\partial \tau _{ij}^R}{\partial x_j}, \end{aligned}$$

(A4)

$$\begin{aligned} \tau _{ij}^R= & {} \overline{U_iU_j} - {\bar{U}}_i{\bar{U}}_j. \end{aligned}$$

(A5)

The closure of LES is achieved by modeling subgrid-scale (SGS) stresses ($\tau _{ij}^R$). The standard isotropic eddy viscosity model, dynamic Smagorinksy [13], with modifications according to Lilly [23] is used. In case of DNS, the SGS stresses in Eqs. (A4)–(A5) are zero.

1.2 Index of quality:

The quality of an LES depends on the size of grid, filter and the chosen subgrid scale (SGS) model. The error in LES is a sum of the numerical discretization error and the modeling (SGS contributions) error. The dependence of these two factors on the grid size makes the estimation of the aggregate error non-trivial [33, 34]. Indices based on TKE are introduced by Celik et al. [6] to give a quantitative comparison between LES and DNS and also to assess the quality of the grid resolution chosen. The true index, IQ_DNS given in Eq. (A6) is used to assess the quality of LES simulation in comparison to DNS. In addition, the index LES_IQ, as defined in Eq. (A7), is used to understand the relative quality of grid size and distribution. It is to be noted that Eq. (A7) is preferable when the grid size is equal to the filter size of LES. An LES grid with corresponding LES_IQ value between 75% and 85% can be considered adequate for engineering applications [6].

$$\begin{aligned} IQ\_DNS = 1- \frac{|k^{DNS}-k^{res}|}{k^{DNS}} \end{aligned}$$

(A6)

$k^{DNS}$ refers to TKE calculated from DNS while $k^{res}$ refers to resolved TKE from LES.

$$\begin{aligned} LES\_IQ = 1 - \frac{|k^{tot}-k^{res}|}{k^{tot}} \end{aligned}$$

(A7)

$k^{tot}$ in Eq. (A7) refers to the total TKE, defined as $k^{tot}$=$k^{res}$+$k^{SGS}$+$k^{num}$, where $k^{SGS}$ and $k^{num}$ refer to the contributions from the SGS model and the numerical dissipation, respectively. $k^{tot}$ is calculated by:

$$\begin{aligned} k^{tot}=k^{res}+a_k(g^p).\quad a_{k} = \frac{1}{g_2^p}\left( \frac{k^{res}_2-k^{res}_1}{\alpha ^{p}-1}\right) .\quad \alpha = g_2/g_1. \end{aligned}$$

(A8)

Based on Richardson’s extrapolation, $a_k$ is expressed as a function of kinetic energy from a finer grid ($k^{res}_2$) and a coarse grid ($k^{res}_1$). The grid size g is given by $g = (\delta x \, \delta y \, \delta z)^{\frac{1}{3}}$, where $\delta x$, $\delta y$ and $\delta z$ refer to the cell dimensions in the x, y and z directions, respectively. Here, p is the order of accuracy of the numerical scheme, in the current study $p=2$. The indices are evaluated on seven grids (Table 1) for ${\text {Re}}_{\tau }= 180$ and four grids (Table 2) for ${\text {Re}}_{\tau }= 400$ (see [38] for more details). Figure 18 represents the IQ_DNS and LES_IQ, along the wall-normal direction, from the chosen grids of LES, viz. Grid7 and Grid1 for the flows at ${\text {Re}}_{\tau }=$ 180 and 400, respectively. The LES_IQ value which is above 85% for ${\text {Re}}_{\tau }=180$ and above 80% for ${\text {Re}}_{\tau }=400$ verifies that sufficient grid resolution is used so that the simulations can qualify as LES. On the other hand, the values of IQ_DNS, being above 85% for both ${\text {Re}}_{\tau }$ cases, validate the accuracy of the simulation.

Table 1 List of LES of rough channel flow for ${\text {Re}}_{\tau } = 180$. $\delta z_{w}^{+}$ and ${\delta }z_{c}^{+}$ indicate the grid spacings near the wall and at the channel centerline, respectively

Table 2 List of LES of rough channel flow for ${\text {Re}}_{\tau } = 400$. ${\delta }z_{w}^{+}$ and ${\delta }z_{c}^{+}$ indicates the grid spacings near the wall and at the channel centerline, respectively

Fig. 18

Indices of quality, a $LES\_IQ$ and b $IQ\_DNS$, for the rough-channel-flow LESs at ${\text {Re}}_{\tau } =$ 180 and 400. Grid7 was chosen for ${\text {Re}}_{\tau }=180$ (Table 1) and Grid1 for ${\text {Re}}_{\tau }=400$ (Table 2)